Novel Information Measures for Fermatean Fuzzy Sets and Their Applications to Pattern Recognition and Medical Diagnosis

Fermatean fuzzy sets (FFSs) have piqued the interest of researchers in a wide range of domains. The striking framework of the FFS is keen to provide the larger preference domain for the modeling of ambiguous information deploying the degrees of membership and nonmembership. Furthermore, FFSs prevail over the theories of intuitionistic fuzzy sets and Pythagorean fuzzy sets owing to their broader space, adjustable parameter, flexible structure, and influential design. The information measures, being a significant part of the literature, are crucial and beneficial tools that are widely applied in decision-making, data mining, medical diagnosis, and pattern recognition. This paper aims to expand the literature on FFSs by proposing many innovative Fermatean fuzzy sets-based information measures, namely, distance measure, similarity measure, entropy measure, and inclusion measure. We investigate the relationship between distance, similarity, entropy, and inclusion measures for FFSs. Another achievement of this research is to establish a systematic transformation of information measures (distance measure, similarity measure, entropy measure, and inclusion measure) for the FFSs. To accomplish this aim, new formulae for information measures of FFSs have been presented. To demonstrate the validity of the measures, we employ them in pattern recognition, building materials, and medical diagnosis. Additionally, a comparison between traditional and novel similarity measures is described in terms of counter-intuitive cases. The findings demonstrate that the innovative information measures do not include any absurd cases.


Introduction
Te idea of the fuzzy set (FS) was developed by Zadeh [1] in 1965, which addressed vagueness and ambiguity in realworld situations. In 1970, Bellman and Zadeh (1970) introduced the concept of decision-making (DM) problems with uncertainty. DM is a systematic procedure of selecting the most ideal choice from a collection of available alternatives. Terefore, the decision maker plays a crucial role in real world environments [2]. A smart decision may have a signifcant impact on the direction of someone's lifestyle.
Before making a fnal selection, a DM assesses the restrictions, advantages, and characteristics of each alternative. Since an FS is defned by a single parameter: membership degree. Several higher-order FSs have been described in recent decades by several scholars.
Atanassov [3] established the notion of intuitionistic fuzzy sets (IFSs) capable of dealing with complexity and uncertainty and it has been extensively examined and utilized by several researchers in DM problems. An IFS is defned by three parameters: membership grade (MG), nonmembership grade (NMG), and hesitancy margin with the property that the sum of MG and NMG must be less than or equal to 1. In many situations, it is conceivable that the sum of the MG and NMG will be greater than 1. To overcome these challenges, Yager [4] introduced the Pythagorean fuzzy set (PyFS) as an extension of the IFS theory. PyFS is defned by an MG and NMG and satisfes the criterion that the square sum of its MG and NMG is less than or equal to 1. Terefore, PyFSs can more accurately express the fuzzy nature of information than IFS.
In the feld of PyFS, there are various approaches for solving real-life multiattribute decision-making (MADM) situations. A number of researchers have also suggested realworld applications in a Pythagorean fuzzy environment. However, if orthopair FSs as 〈0.9, 0.5〉, where 0.9 is the MG of specifc criteria of a parameter and 0.5 is the NMG, it does not fulfll the IFS and PFS requirements. However, the cubic sum of the MG and NMG is equal to or less than one. In this context, Senapati and Yager [5] recently introduced the Fermatean fuzzy set (FFS). Tey also demonstrated that FFSs have larger degrees of uncertainty than IFSs and PyFSs, are capable of sustaining higher levels of uncertainty, and can solve MCDM challenges. Information measures are an essential notion for dealing with MADM challenges in a variety of domains, including pattern recognition, clinical diagnosis, and personnel appointment. Tere are several types of information measures established such as distance, similarity, entropy, and inclusion measures.
Te MADM process are normally assisted by similarity measures, distance measures, inclusion measures, entropy measures, and, in certain situations, aggregation operators. Te degree of similarity measures has garnered considerable interest in recent decades due to its importance in DM, data mining, pattern recognition, and medical diagnosis applications. Szmidt and Kacprzyk [6] performed the frst investigation, extending well-known distance measures such as the Hamming distance and the Euclidian distance to the IFS environment and comparing them to approaches used for conventional fuzzy sets. However, Wang and Xin [7] suggested that Szmidt and Kacprzyk [6] distance measure was inefective in certain situations. Terefore, several innovative pattern recognition distance measures were developed and implemented. Grzegorzewski [8] also extended Hamming, Euclidean, and their normalized versions to the IFS framework. Later on, Chen [9] demonstrated that several faws occurred in Grzegorzewski [8] by providing counterexamples. Hung and Yang [10] described three similarity measures and extended the Hausdorf distance to IFSs. On the other side, rather than expanding well-established measures, various research established novel similarity measures for IFS.
Yong et al. [11] developed a novel similarity measure for IFS based on MG and NMG. Mitchell [12] demonstrated that Yong et al.'s [11] similarity measure had certain counter-intuitive circumstances and improved it statistically. Additionally, Liang and Shi [13] provided examples to demonstrate that the similarity measure proposed by Yong et al. [11] was unsuitable for certain scenarios, and hence developed various additional similarity measures for IFSs.
Xu [14] formulated a series of IFS-based similarity measures and applied them to the MADM problem employing IF information. Xu and Chen [15] presented a set of distance and similarity measures that are diferent combinations and extensions of the weighted Hamming, Euclidean, and Hausdorf distances. Xu and Yager [16] constructed a similarity measure between IFSs and used it to MAGDM utilizing IF preference relations.
In addition to this research, several researchers investigated the relationships between IFSs' distance, similarity, and entropy measures. Zeng and Guo [17] analyzed the relationship between normalized distance, similarity, inclusion, and entropy of interval-valued fuzzy collections. Additionally, it was demonstrated that the similarity, inclusion, and entropy of interval-valued fuzzy sets may be induced using the normalized distance of their axiomatic defnitions. Wei et al. [18] proposed a generalized entropy measure for IFSs and PyFSs. Additionally, a technique was developed for constructing similarity measures for IFS and PyFSs using entropy measures. Numerous researchers investigated information measures (distance measure, similarity measure, entropy measure, and inclusion measure) for IFSs and PyFSs and their transformations relationship. Dengfeng and Chuntian [19] investigated the similarity between IFSs and used their fndings to pattern recognition. Huang and Yang [10] presented the Hausdorf distance as a similarity measure between IFSs and utilized it to assess the degree of similarity between IFSs. Ashraf et al. [20] gave the idea of a spherical fuzzy set then they implicated this concept also in decision-making [21].
Nguyen et al. [22] developed a novel knowledge-based similarity measure for IFSs and demonstrated its application to pattern recognition. Zhang [23] pioneered a unique strategy for PyFSs MADM based on similarity measures. Zhang et al. [24] explored the use of the application of a scoring function on IFSs with double parameters for pattern recognition and medical diagnosis. Ejegwa established distance [25] and similarity measures [26] for PyFSs.
Ye [27] designed and implemented a cosine similarity measure for IFSs (CIFS). In addition, Ye [28] introduced the cosine similarity measure for interval-valued IFSs (CIVIFSs) and described its use in solving MADM problems. Liu et al. [29] investigated the cosine similarity measure between hybrid IFSs and their application for diagnostic purposes. In recent years, several scholars have conducted research on PyFS information measures (distance measure, similarity measure, entropy measure, and inclusion measure). Wei and Wei [30] introduced a set of ten cosine-based PFS similarity measures relying on the MG, NMG, and hesitation of PyFSs in order to improve the capacity to cope with the two optimization challenges related to pattern recognition and medical diagnosis procedures. Peng [31] established a PyFS similarity measure based on the parameters Lp norm and levels of ambiguity, which were examined in detail in relation to the PyFS similarity measure. Peng et al. [32] developed the fundamental defnitions of PyFS information measures, along with the similarity measure, as well as discussed the transformation principles for the established information measures. 2 Computational Intelligence and Neuroscience Te advantages of existing information measures are as follows: An examination of the existing literature on FS, IFS, and PyFS exposes a number of weaknesses that spur us to create a more potent class of novel information measures (distance measure, similarity measure, entropy measure, and inclusion measure).
Te disadvantages of existing information measures are as follows: (i) Some of them cannot help but be caught in pointless circumstances (i.e., dividing by zero). (ii) Many of them struggle to avoid examples that seem to go against logic.
(iii) Many of them are unable to categorize the results, and some of them provide irrational results. We describe a class of useful FFS information measures (distance measure, similarity measure, entropy measure, and inclusion measure), ofer associated information measure formulations, and examine their transformation connections to address the faw in the prior research.
Te important contributions of the current manuscript are listed. Te manuscript is organized as follows: Section 2 discusses the defnitions and fundamental ideas of FS, HFS, IFS, and FFS, as well as the corresponding operational rules of FFS. Section 3 introduces a new type of information measures, provides related information measure formulations, and investigates their transformation relationships for FFSs. In Sections 4 to 7, we demonstrated the application of the novel information measures between FFSs to pattern recognition. Moreover, a comparative study has been presented between the proposed similarity measure and conventional similarity measures. Section 7 concludes the paper by outlining the future area of research.

Basic Terminologies
In this section, we provide some relevant fundamental information, such as FS, HFS, IFS, FFSs, and some related operational laws, which are listed. Tese core concepts will assist readers in comprehending the proposed framework.

Some New Types of Information Measures between FFSs
Tis section explains the axiomatic framework of FFSs information measures (distance, similarity, entropy, and inclusion), as well as their related formulations. Simultaneously, their transformation relationships are thoroughly examined.

Distance Measures for FFSs.
Tis section introduces the idea of a distance measures for FFSs. A number that is assigned to a pair of points in a space which indicates how far those points are from one another. A distance measure is called a metric if it is always positive and also it is always symmetric. (1) 0 ≤ D(Y, P) ≤ 1; (2) D(Y, P) � D(P, Y); Computational Intelligence and Neuroscience Similarity Measure for FFSs. Tis section introduces the idea of similarity measures for FFSs. Similarity functions take a pair of points and return a large similarity value for nearby points, and a small similarity value for distant points. One way to transform between a distance function and a similarity measure is to take the reciprocal.

Theorem 2. Let Y and P be two FFSs, then
Theorem 5. For i � 1, 4, 5, 6 and for all ϰ i ∈ Z, (I 3 3.3. Entropy for FFSs. Let Y and P two FFSs on Z. An carrying the following features: is an inclusion measure.

Te Relations between Tese Measures.
In this section, we study the relations between inclusion, entropy, similarity measure, and distance measure of Fermatean fuzzy sets. First, according to the defnitions of similarity measure and distance measure of Fermatean fuzzy sets, one should note that they are all used for estimating the degree of similarity between two Fermatean fuzzy sets. Te main diference is as follows: for the similarity measure, a greater value means that the two Fermatean fuzzy sets are more similar than are a pair with a lower value. Te situation for the distance measure is just the opposite, that is, the smaller the value is, the more similar these two Fermatean fuzzy sets are. So, we can obtain the following theorem.

Transformation Relationships among Information Measures for FFSs
Theorem 8. Suppose D be the Fermatean fuzzy distance measure for Y, P ∈ FFS s, then S(Y, P) � 1 − D(Y, P) is the similarity measure of FFS s Y and P. Te proof is straightforward.
Theorem 10. Let D and S be the distance and similarity measures of Tis completes the proof. □ Theorem 11. For Y, P ∈ FFSs, and we order Computational Intelligence and Neuroscience Theorem 12. Let D be the distance measure and S be the similarity measure of FFSs, for Y in FFSs, then Tis completes the proof.

Proof
(i) (I 1 ) It is straightforward. Defnition 11. Let Y and P be two FFS s, then we defne g(A,B) ∈ FFS s, ∀ϰ i ∈ Z,

Theorem 15.
Suppose E be the entropy measure of FFSs, for Y, P ∈ FFSs, then E(g(Y, P)) is the similarity measure of FFSs Y and P. Proof Also, we can know that is Computational Intelligence and Neuroscience and known by the defnition, E(g(Y, O)) ≤ E(g (Y, P)).
Similarly, we can prove that E(g(Y, O)) ≤ E(g (P, O)). Tis completes the proof. □ □ Theorem 16. Suppose I be the inclusion measure of FFS s, According to the defnition of inclusion measure, we have , that is E(A) ≤E(P). Tis completes the proof. □ Example 1. For Y, P ∈ FFS s, and we order I(Y, P) � I 1 (Y, P) Known by the defnition of similarity measure of FFSs, we have According to the defnition of similarity measure, we can have where L is an unknown pattern. Its aimed is to determine the class to which L belongs. In order to do that, the distance between L and classes M 1 , M 2 , M 3 , and M 4 are measured, and L is then allocated to the class M g specifed as follows: For all the newly developed distance measures (D 1 −  Table 1. It is observed in the Table 1, that an unknown pattern L belongs to a class M 3 when D 1 to D 13 are used. It is clear that the cause for this diference is the frst characteristic, i.e., (ϰ 1 ). Te FFNs of ϰ 1 are as follows: is more acceptable. By routine calculations, we can fnd the aforementioned relation for D to D 13 as shown in Table 1.

Example 3.
Assume that a doctor would like to diagnose the condition of C: (viral fever, malaria, typhoid, or chest problem) for patients P: (Ragu, Mathi,Velu, and Karthi) with disease symptoms V: (headache, acidity, burning eyes and depression). Te symptoms associated with the considered diagnosis are listed in Table 2-4, and the symptoms of the disease associated with each patient are listed in Table 2. Each table element is represented by a specifc FFSs. For each patient, a precise diagnosis is necessary.Te distance measuring methods mentioned here are used to assess the distance between each patient and each diagnosis. Each patient was then diagnosed using the concept of the shortest possible distance. To determine a condition of the patient, we may assess the distance measure between the symptoms associated with each illness and those associated with the patient. Te diagnostic fndings are provided in Table 2-4 using the distance measure formula D 13 . We may conclude that all the patients sufer from viral fever.

Comparison of the Distance Measure between FFSs in Medical Diagnosis.
To illustrate the efectiveness of the novel distance measure for specifc FFSs in pattern recognition, we present a numerical example and compare the novel fndings to those reported in the literature.  Table 5. Te FF relation S ⟶ L is denoted by the FFS, as seen in Table 6. Each element in Table 6 is represented by FFS. Te established distance measure methods are used to determine the distance between each patient and each diagnosis. Ten, using the idea of minimal distance degree, each patient was diagnosed. We demonstrated the distance measure results of the patient M j (j � 1, 2, 3, 4) with regard to the diagnostic L i (i � 1, 2, 3, 4, 5) and the fnal diagnosis fndings are given in Al has malaria Table 7, Bob has stomach problem Table 8, Joe has typhoid Table 9, and Ted has viral fever Table 10. We perform a comparison study with other methodologies to demonstrate the capability and validity of the presented distance measures, and the fndings are provided in Table 11. Table 11 shows that the suggested distance measure approaches achieve the same result as in D [18], D [35], D [36], and D [37], demonstrating that using the proposed distance measure methods to solve the medical diagnosis problem is possible and benefcial. From the preceding practical implementation of the measures techniques, we may deduce that the proposed distance measures approaches are more efective and superior in handling real world challenges.

Apply the Similarity Measure between FFSs to Pattern Recognition
In this part, we describe some examples to show the use of the suggested similarity measures based on FFS to pattern recognition.
Example 5. Suppose the four classes M 1 , M 2 , M 3 , and M 4 of known construction materials and L, an unknown construction material, are defned in the space X � ϰ 1 , ϰ 2 , ϰ 3 and are represented by FFS is given. Its goal is to ascertain to which class L belongs to (see Table 12).
Here, L is a known building materials. Its objective is to determine the class to which L belongs. To do this, the degrees of similarity between L and classes M 1 , M 2 , M 3 , and M 4 are measured, and L is then allocated to the class M g specifed as follows:  Table 13. It is clearly observed in the Table 13, that an unknown building material L belongs to a class M 1 when S 1 , S 3 , S 7 , to S 10 are used and L belongs to a class M 3 when S 2 , S 4 , to S 6 and S 11 to S 13 are used. It is clear that the cause for this diference is the frst feature, i.e., (ϰ 1 ). Te

A Comparison of the Proposed Similarity Measures between FFSs
To illustrate the efectiveness of the novel similarity measures for specifc FFSs in pattern recognition, we present some examples and compare the novel fndings to those reported in the literature. Our objective is to ascertain the class to which L belongs. Te classifcation result of the suggested similarity measures (S 1 − S 13 ) displayed in Table 14  Example 7. Assume that a doctor would like to diagnose the condition of C: (viral fever, malaria, or typhoid) for a set of patients P: (Al, Bob, Joe, and Ted) having symptoms V: (temperature, headache, and cough). Te symptoms associated with the considered diagnosis are listed in Table 16, and the symptoms associated with each patient are listed in Table 17. Each table element is represented by a specifc FFSs. Each patient requires proper diagnosis, which need be assessed. We will identify a diagnosis for each patient based on the similarity between the symptoms associated with each diagnosis and those associated with the patient. Te diagnostic observations are described in Table 18 Al, Table 19 Bob, Table 20 Joe, and Table 21 Ted, respectively, using the novel similarity measures formula (S 1 − S 13 ). Te patient Al is diagnosed with malaria (Mal.) in 12 of the 13 of the approaches; the remaining approach indicates that Al is diagnosed with viral fever (VF) as presented in Table 18. It is obvious that Bob has a stomach problem (SP), since all of the measures yield the same fndings as shown in Table 19. Joe is diagnosed with typhoid in 12 of the 13 methods; the other approach represented that Joe is diagnosed with VF as shown in Table 20. Similarly, 9 of Table 1: Distance measures for Example 2 with α � β � 0.5, l 1 � l 2 � 2, and t � 1.    the 13 measures indicated that Ted has VF, whereas, the remaining methods imply that Ted has Mal as presented in Table 21. For patient Al, it could be observed from Table 18 and Table 22 50]. For patient Bob the novel similarity measures provided the same results as in the literature presented in Table 19 and  Table 22. Similarly, for patient Joe the proposed similarity measure provided the same result as in the literature shown in Table 20 Tables 21 and 22. Table 23 shows the present summary of medical diagnosis.

Application of the Inclusion Measure between FFSs and Pattern Recognition
Tis section illustrates the applicability of the suggested FFS inclusion measures to pattern recognition.  Table 24. It is clearly observed in the Table 24 that an unknown pattern L belongs to a class M 4 when I 1 to I 3 and I 5 to I 6 are used, and L belongs to a class M 3 and M 2 when I 4 and I 7 are, respectively, used. It is clear that the cause for this diference is the frst characteristic, i.e., (ϰ 1 ). Te FFNs of ϰ 1 are is more acceptable. In a similar way, we can fnd the previously mentioned relations for I 2 to I 7 .               (1) We developed axiomatically FFSs information measures (distance measure, similarity measure, entropy, and inclusion measure). (2) We constructed various formulae for FFSs information measures and analyzed the associated transformation relationships in detail. (3) We used the established distance measures (D 1 -D 13 ) to pattern recognition and medical diagnosis to demonstrate their efcacy. Te applications substantiate the results and also illustrate the feasibility and efectiveness of the distance measures between FFSs information. (4) We demonstrated the efcacy of the novel similarity measures (S 1 -S 13 ); several counterintuitive examples of existing similarity measures are shown. We employed them to pattern recognition, construction materials, and medical diagnosis. For pattern recognition problems, we conclude that the proposed similarity measures dominate existing similarity measures. In some special situations, it has been shown that many conventional similarity measures are incapable of providing reasonable fndings. However, in these specifc cases, the proposed similarity measure is profcient of discriminating Te experimental fndings demonstrated that the proposed measures are more reliable and can avoid the counter-intuitive situation in dealing with practical applications based on Fermatean fuzzy environment. [24,51,52].

Limitations and Future Works
(1) Te FFSs are inappropriate to deal with situations where the cube sum of membership and nonmembership grades of exceeds 1 (2) A near future target is to unfold the application of the proposed information measures in scientifc investigations for decision-making, pattern recognition, linguistic summarization, and data mining (3) We have also a plan to apply the presented approach to procurement planning, water desalination station  selection, wind power plant site selection, and many more domains of real world problems (4) Additionally, we will be further interested to immerse them in a variety of fuzzy environments (5) Furthermore, since this work presents an applicative analysis of the FFS information measures, we should develop an appropriate software to efectively apply the presented information measures in a realistic situation

Data Availability
Te data used in this manuscript are hypothetical and can be used by anyone by just citing this article.